how do you mute a note as part of a chord on guitar?

If you want to avoid hitting that lowest string, aim more accurately with your picking hand. Muting strings in the middle is one thing, but if you are picking correctly (finger style or with a pick) you shouldn't be hitting that string if you don't want to. It may help to (when playing pick style) lightly rest the heel of your picking hand somewhat on that lowest string and near the bridge.

Edit: Thanks for the clarification. I'd say that your playing hand index finger fretting that F (G string 10th fret in your example) is in a great position to just lightly rest on that string (the meaty part of the tip of your index finger) and thus mute it. There will be slightly more string noise depending on your sound (electric, acoustic, settings, etc.) but it should be fine.

There is a slight difference in sound between a muted string and a non-played string, but most times you see tab like that it's because you want the middle string to be silent. If you truly want to mute the string, fret the strings with your 1st and 3rd finger and mute the middle string with your 2nd finger.

If I were to play that fingering, I'd forget trying to mute the middle string and simply not play it. The way you do this is to use one of your free fingers to finger-pick the higher string while the pick plucks the lower string. You're holding the pick with your thumb and forefinger....that leaves 3 other fingers just going along for the ride. You might as well put them to work, right? Here's a video I made which shows this technique: http://www.youtube.com/watch?v=4HEHvgPmSaA

Well a pretty easy way is to just take the finger you're playing on the 10 and have it slightly rest on the c string and that should mute it pretty well :) Just mess around with it a bit.. I have never taken any guitar lessons, but I've learned the guitar myself and I perform at local wineries. So this technique worked for me when I was playing electric, I've since moved onto a 12-string acoustic.

you use the playing hand.just rest your finger on the C string. That would be called a dead note.